# How do you calculate the change in temperature of an adiabatic system? [closed]

I really don't even know how to start doing this problem.

I understand PV=nRT and I understand that if either T, V, or P is kept constant, then the other 2 change at the same rate, and I understand that if either T, V, or P is kept constant while the other two change, then the system experiences a change in heat. I also understand that the internal energy is kept constant and that internal energy = heat - work. However I do not know how to relate the change in temperature to a change in heat. Can somebody please walk me through this?

Here is a diagram to better visualize the problem.

My homework question asks to find the temperature at point c

I have already calculated that n=0.41 and that the pressure at point b is 2.01966 atm

You do not need to solve for c, just please tell me how.

Okay so I couldn't handle it anymore and I just looked at the answer which happened to be 810J. This make me even more confused. Where did the extra 110J of energy just appear from?

• Use the notion that, since the process is adiabatic, $U_a=U_c$.
– xish
Dec 11, 2013 at 8:21
• I know that, but that doesn't give me a temperature. Dec 11, 2013 at 8:34
• Also, it looks like my biggest problem here is actually finding the work done on the gas between a and b Dec 11, 2013 at 8:34
• Is it an ideal gas? In that case, it's trivial. Jan 10, 2014 at 14:53
• Are you given whether the gas in monoatomic or diatomic or...? May 15, 2014 at 4:11

An ideal gas undergoing an adiabatic transformation is subjected to the law:

$$P\,V^{\gamma} = \operatorname{constant}$$ where $$\gamma$$ is a nice fraction that depends on the number of atoms in the gas molecule. For a monatomic gas (like helium) $$\gamma = 5/3$$, for a diatomic gas (like hydrogen, oxygen, nitrogen...) and linear molecules (like carbon dioxide) $$\gamma = 7/5$$, in all the other cases is $$4/3$$ (but then the ideal gas hypothesis becomes weak).

This equation will allow you to find out $$P_c$$ starting from the point $$a$$. Then you can use the equation of status, which of course is still valid, to work out also $$T_c$$.

As long as you are familiar with the equation $W=-\int P\ dV$, this should be pretty straightforward. If you aren't, there should be a derivation for this formula in your text.

The ideal gas law states:

$$PV=nRT \rightarrow P=\frac{nRT}{V}$$

We know that $n$ and $R$ are constant and that the temperature, $T$, is constant because the process between $a$ and $b$ is isothermal. Now that we have the pressure as a function of volume, it's pretty straightforward to use this in conjunction with the above work equation and integrate from the initial volume to the final volume to find the work done between states $a$ and $b$.

It seems like you're aware of how to continue the problem after this point.

• Actually, I got that part while I was waiting for an answer, I don't know how to find the heat input from point b to c now. Dec 11, 2013 at 10:38
• The principle is the same. $W=-\int P\ dV$. The answer is a bit more trivial, though.
– xish
Dec 11, 2013 at 10:44
• Which pressure should I be using? Dec 11, 2013 at 10:55
• Hint: The pressure along the process is irrelevant. What are the bounds of the integral?
– xish
Dec 11, 2013 at 10:57
• The bounds of the integral are vb and va. Also, the formula to calculate work doesn't really help me because I am trying to calculate the transfer of heat. I understand that the work is the same. That work is 706J, but I don't know how to turn that 706J into kelvins Dec 11, 2013 at 11:01